i would remove the broken stud by using a sturdy "pitman arm puller" type of tool.
provided it can be fitted, crank it up tight on the stud, apply some heat, then a sharp wack of a hammer.
repeat as necessary.
i have only had to do this a few times over the almost 60 years playing with our "toys".
a couple of these came loose surprisingly easy, while a few were real "fun".
the one thing you don't have to worry about is boogering up the threads.
OP, you can do this. it's really not as hard as it seems.
please let us know how you make out.